Centroid homomorphisms

Let A be a (non unital, non associative) algebra. The centroid of A is the set of linear maps T on A such that T commutes with left and right multiplication, that is to say, for all a and b in A, $$ T(ab) = (Ta)b, T(ab) = a(Tb). $$ In mathlib we call elements of the centroid "centroid homomorphisms" (CentroidHom) in keeping with AddMonoidHom etc.

We use the FunLike design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types.

Types of morphisms

• CentroidHom: Maps which preserve left and right multiplication.

Typeclasses

• CentroidHomClass

References

• [Jacobson, Structure of Rings][Jacobson1956]
• [McCrimmon, A taste of Jordan algebras][mccrimmon2004]

Tags

centroid

structure CentroidHom (α : Type u_6) [NonUnitalNonAssocSemiring α] extends AddMonoidHom : Type u_6

The type of centroid homomorphisms from α to α.

• toFun : α → α
• map_zero' : ZeroHom.toFun (↑s.toAddMonoidHom) 0 = 0
• map_add' : ∀ (x y : α), ZeroHom.toFun (↑s.toAddMonoidHom) (x + y) = ZeroHom.toFun (↑s.toAddMonoidHom) x + ZeroHom.toFun (↑s.toAddMonoidHom) y
• map_mul_left' : ∀ (a b : α), ZeroHom.toFun (↑s.toAddMonoidHom) (a * b) = a * ZeroHom.toFun (↑s.toAddMonoidHom) b

  Commutativity of centroid homomorphims with left multiplication.

• map_mul_right' : ∀ (a b : α), ZeroHom.toFun (↑s.toAddMonoidHom) (a * b) = ZeroHom.toFun (↑s.toAddMonoidHom) a * b

  Commutativity of centroid homomorphims with right multiplication.

class CentroidHomClass (F : Type u_6) (α : outParam (Type u_7)) [NonUnitalNonAssocSemiring α] extends AddMonoidHomClass : Type (max u_6 u_7)

CentroidHomClass F α states that F is a type of centroid homomorphisms.

You should extend this class when you extend CentroidHom.

• coe : F → α → α
• coe_injective' : Function.Injective FunLike.coe
• map_add : ∀ (f : F) (x y : α), f (x + y) = f x + f y
• map_zero : ∀ (f : F), f 0 = 0
• map_mul_left : ∀ (f : F) (a b : α), f (a * b) = a * f b

  Commutativity of centroid homomorphims with left multiplication.

• map_mul_right : ∀ (f : F) (a b : α), f (a * b) = f a * b

  Commutativity of centroid homomorphims with right multiplication.

instance instCoeTCCentroidHom {F : Type u_1} {α : Type u_5} [NonUnitalNonAssocSemiring α] [CentroidHomClass F α] : CoeTC F (CentroidHom α)

Equations

• One or more equations did not get rendered due to their size.

Centroid homomorphisms

instance CentroidHom.instCentroidHomClassCentroidHom {α : Type u_5} [NonUnitalNonAssocSemiring α] : CentroidHomClass (CentroidHom α) α

Equations

• One or more equations did not get rendered due to their size.

instance CentroidHom.instCoeFunCentroidHomForAll {α : Type u_5} [NonUnitalNonAssocSemiring α] : CoeFun (CentroidHom α) fun (x : CentroidHom α) => α → α

Helper instance for when there's too many metavariables to apply FunLike.CoeFun directly.

Equations

• CentroidHom.instCoeFunCentroidHomForAll = inferInstanceAs (CoeFun (CentroidHom α) fun (x : CentroidHom α) => α → α)

theorem CentroidHom.toFun_eq_coe {α : Type u_5} [NonUnitalNonAssocSemiring α] {f : CentroidHom α} : f.toFun = ⇑f

theorem CentroidHom.ext {α : Type u_5} [NonUnitalNonAssocSemiring α] {f : CentroidHom α} {g : CentroidHom α} (h : ∀ (a : α), f a = g a) : f = g

@[simp]

theorem CentroidHom.coe_toAddMonoidHom {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) : ⇑↑f = ⇑f

@[simp]

theorem CentroidHom.toAddMonoidHom_eq_coe {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) : f.toAddMonoidHom = ↑f

theorem CentroidHom.coe_toAddMonoidHom_injective {α : Type u_5} [NonUnitalNonAssocSemiring α] : Function.Injective AddMonoidHomClass.toAddMonoidHom

def CentroidHom.toEnd {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) : AddMonoid.End α

Turn a centroid homomorphism into an additive monoid endomorphism.

Equations

• CentroidHom.toEnd f = ↑f

theorem CentroidHom.toEnd_injective {α : Type u_5} [NonUnitalNonAssocSemiring α] : Function.Injective CentroidHom.toEnd

def CentroidHom.copy {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) (f' : α → α) (h : f' = ⇑f) : CentroidHom α

Copy of a CentroidHom with a new toFun equal to the old one. Useful to fix definitional equalities.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CentroidHom.coe_copy {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) (f' : α → α) (h : f' = ⇑f) : ⇑(CentroidHom.copy f f' h) = f'

theorem CentroidHom.copy_eq {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) (f' : α → α) (h : f' = ⇑f) : CentroidHom.copy f f' h = f

def CentroidHom.id (α : Type u_5) [NonUnitalNonAssocSemiring α] : CentroidHom α

id as a CentroidHom.

Equations

• One or more equations did not get rendered due to their size.

instance CentroidHom.instInhabitedCentroidHom (α : Type u_5) [NonUnitalNonAssocSemiring α] : Inhabited (CentroidHom α)

Equations

• CentroidHom.instInhabitedCentroidHom α = { default := CentroidHom.id α }

@[simp]

theorem CentroidHom.coe_id (α : Type u_5) [NonUnitalNonAssocSemiring α] : ⇑(CentroidHom.id α) = id

@[simp]

theorem CentroidHom.toAddMonoidHom_id (α : Type u_5) [NonUnitalNonAssocSemiring α] : ↑(CentroidHom.id α) = AddMonoidHom.id α

@[simp]

theorem CentroidHom.id_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (a : α) : (CentroidHom.id α) a = a

def CentroidHom.comp {α : Type u_5} [NonUnitalNonAssocSemiring α] (g : CentroidHom α) (f : CentroidHom α) : CentroidHom α

Composition of CentroidHoms as a CentroidHom.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CentroidHom.coe_comp {α : Type u_5} [NonUnitalNonAssocSemiring α] (g : CentroidHom α) (f : CentroidHom α) : ⇑(CentroidHom.comp g f) = ⇑g ∘ ⇑f

@[simp]

theorem CentroidHom.comp_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (g : CentroidHom α) (f : CentroidHom α) (a : α) : (CentroidHom.comp g f) a = g (f a)

@[simp]

theorem CentroidHom.coe_comp_addMonoidHom {α : Type u_5} [NonUnitalNonAssocSemiring α] (g : CentroidHom α) (f : CentroidHom α) : ↑(CentroidHom.comp g f) = AddMonoidHom.comp ↑g ↑f

@[simp]

theorem CentroidHom.comp_assoc {α : Type u_5} [NonUnitalNonAssocSemiring α] (h : CentroidHom α) (g : CentroidHom α) (f : CentroidHom α) : CentroidHom.comp (CentroidHom.comp h g) f = CentroidHom.comp h (CentroidHom.comp g f)

@[simp]

theorem CentroidHom.comp_id {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) : CentroidHom.comp f (CentroidHom.id α) = f

@[simp]

theorem CentroidHom.id_comp {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) : CentroidHom.comp (CentroidHom.id α) f = f

@[simp]

theorem CentroidHom.cancel_right {α : Type u_5} [NonUnitalNonAssocSemiring α] {g₁ : CentroidHom α} {g₂ : CentroidHom α} {f : CentroidHom α} (hf : Function.Surjective ⇑f) : CentroidHom.comp g₁ f = CentroidHom.comp g₂ f ↔ g₁ = g₂

@[simp]

theorem CentroidHom.cancel_left {α : Type u_5} [NonUnitalNonAssocSemiring α] {g : CentroidHom α} {f₁ : CentroidHom α} {f₂ : CentroidHom α} (hg : Function.Injective ⇑g) : CentroidHom.comp g f₁ = CentroidHom.comp g f₂ ↔ f₁ = f₂

instance CentroidHom.instZeroCentroidHom {α : Type u_5} [NonUnitalNonAssocSemiring α] : Zero (CentroidHom α)

Equations

• One or more equations did not get rendered due to their size.

instance CentroidHom.instOneCentroidHom {α : Type u_5} [NonUnitalNonAssocSemiring α] : One (CentroidHom α)

Equations

• CentroidHom.instOneCentroidHom = { one := CentroidHom.id α }

instance CentroidHom.instAddCentroidHom {α : Type u_5} [NonUnitalNonAssocSemiring α] : Add (CentroidHom α)

Equations

• One or more equations did not get rendered due to their size.

instance CentroidHom.instMulCentroidHom {α : Type u_5} [NonUnitalNonAssocSemiring α] : Mul (CentroidHom α)

Equations

• CentroidHom.instMulCentroidHom = { mul := CentroidHom.comp }

instance CentroidHom.instSMul {M : Type u_2} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] : SMul M (CentroidHom α)

Equations

• One or more equations did not get rendered due to their size.

instance CentroidHom.instIsScalarTowerCentroidHomInstSMulInstSMul {M : Type u_2} {N : Type u_3} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [Monoid N] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] [DistribMulAction N α] [SMulCommClass N α α] [IsScalarTower N α α] [SMul M N] [IsScalarTower M N α] : IsScalarTower M N (CentroidHom α)

Equations

• (_ : IsScalarTower M N (CentroidHom α)) = (_ : IsScalarTower M N (CentroidHom α))

instance CentroidHom.instSMulCommClassCentroidHomInstSMulInstSMul {M : Type u_2} {N : Type u_3} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [Monoid N] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] [DistribMulAction N α] [SMulCommClass N α α] [IsScalarTower N α α] [SMulCommClass M N α] : SMulCommClass M N (CentroidHom α)

Equations

• (_ : SMulCommClass M N (CentroidHom α)) = (_ : SMulCommClass M N (CentroidHom α))

instance CentroidHom.instIsCentralScalarCentroidHomInstSMulMulOppositeMonoidOp_leftToSMulToZeroToMulZeroClassToSMulZeroClassToAddZeroClassToAddMonoidToAddCommMonoidToDistribSMulToSMulToSMulZeroClassToSMulWithZeroOp_left {M : Type u_2} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] [DistribMulAction Mᵐᵒᵖ α] [IsCentralScalar M α] : IsCentralScalar M (CentroidHom α)

Equations

• (_ : IsCentralScalar M (CentroidHom α)) = (_ : IsCentralScalar M (CentroidHom α))

instance CentroidHom.isScalarTowerRight {M : Type u_2} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] : IsScalarTower M (CentroidHom α) (CentroidHom α)

Equations

• (_ : IsScalarTower M (CentroidHom α) (CentroidHom α)) = (_ : IsScalarTower M (CentroidHom α) (CentroidHom α))

instance CentroidHom.hasNpowNat {α : Type u_5} [NonUnitalNonAssocSemiring α] : Pow (CentroidHom α) ℕ

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CentroidHom.coe_zero {α : Type u_5} [NonUnitalNonAssocSemiring α] : ⇑0 = 0

@[simp]

theorem CentroidHom.coe_one {α : Type u_5} [NonUnitalNonAssocSemiring α] : ⇑1 = id

@[simp]

theorem CentroidHom.coe_add {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) (g : CentroidHom α) : ⇑(f + g) = ⇑f + ⇑g

@[simp]

theorem CentroidHom.coe_mul {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) (g : CentroidHom α) : ⇑(f * g) = ⇑f ∘ ⇑g

@[simp]

theorem CentroidHom.coe_smul {M : Type u_2} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] (n : M) (f : CentroidHom α) : ⇑(n • f) = n • ⇑f

@[simp]

theorem CentroidHom.zero_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (a : α) : 0 a = 0

@[simp]

theorem CentroidHom.one_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (a : α) : 1 a = a

@[simp]

theorem CentroidHom.add_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) (g : CentroidHom α) (a : α) : (f + g) a = f a + g a

@[simp]

theorem CentroidHom.mul_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (f : CentroidHom α) (g : CentroidHom α) (a : α) : (f * g) a = f (g a)

@[simp]

theorem CentroidHom.smul_apply {M : Type u_2} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] (n : M) (f : CentroidHom α) (a : α) : (n • f) a = n • f a

@[simp]

theorem CentroidHom.toEnd_zero {α : Type u_5} [NonUnitalNonAssocSemiring α] : CentroidHom.toEnd 0 = 0

@[simp]

theorem CentroidHom.toEnd_add {α : Type u_5} [NonUnitalNonAssocSemiring α] (x : CentroidHom α) (y : CentroidHom α) : CentroidHom.toEnd (x + y) = CentroidHom.toEnd x + CentroidHom.toEnd y

theorem CentroidHom.toEnd_smul {M : Type u_2} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] (m : M) (x : CentroidHom α) : CentroidHom.toEnd (m • x) = m • CentroidHom.toEnd x

instance CentroidHom.instAddCommMonoidCentroidHom {α : Type u_5} [NonUnitalNonAssocSemiring α] : AddCommMonoid (CentroidHom α)

Equations

• One or more equations did not get rendered due to their size.

instance CentroidHom.instNatCastCentroidHom {α : Type u_5} [NonUnitalNonAssocSemiring α] : NatCast (CentroidHom α)

Equations

• CentroidHom.instNatCastCentroidHom = { natCast := fun (n : ℕ) => n • 1 }

@[simp]

theorem CentroidHom.coe_nat_cast {α : Type u_5} [NonUnitalNonAssocSemiring α] (n : ℕ) : ⇑↑n = n • ⇑(CentroidHom.id α)

theorem CentroidHom.nat_cast_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (n : ℕ) (m : α) : ↑n m = n • m

@[simp]

theorem CentroidHom.toEnd_one {α : Type u_5} [NonUnitalNonAssocSemiring α] : CentroidHom.toEnd 1 = 1

@[simp]

theorem CentroidHom.toEnd_mul {α : Type u_5} [NonUnitalNonAssocSemiring α] (x : CentroidHom α) (y : CentroidHom α) : CentroidHom.toEnd (x * y) = CentroidHom.toEnd x * CentroidHom.toEnd y

@[simp]

theorem CentroidHom.toEnd_pow {α : Type u_5} [NonUnitalNonAssocSemiring α] (x : CentroidHom α) (n : ℕ) : CentroidHom.toEnd (x ^ n) = CentroidHom.toEnd x ^ n

@[simp]

theorem CentroidHom.toEnd_nat_cast {α : Type u_5} [NonUnitalNonAssocSemiring α] (n : ℕ) : CentroidHom.toEnd ↑n = ↑n

instance CentroidHom.instSemiringCentroidHom {α : Type u_5} [NonUnitalNonAssocSemiring α] : Semiring (CentroidHom α)

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CentroidHom.toEndRingHom_apply (α : Type u_5) [NonUnitalNonAssocSemiring α] (f : CentroidHom α) : (CentroidHom.toEndRingHom α) f = CentroidHom.toEnd f

def CentroidHom.toEndRingHom (α : Type u_5) [NonUnitalNonAssocSemiring α] : CentroidHom α →+* AddMonoid.End α

CentroidHom.toEnd as a RingHom.

Equations

• One or more equations did not get rendered due to their size.

theorem CentroidHom.comp_mul_comm {α : Type u_5} [NonUnitalNonAssocSemiring α] (T : CentroidHom α) (S : CentroidHom α) (a : α) (b : α) : (⇑T ∘ ⇑S) (a * b) = (⇑S ∘ ⇑T) (a * b)

instance CentroidHom.instDistribMulActionCentroidHomToAddMonoidToAddMonoidWithOneToAddCommMonoidWithOneToNonAssocSemiringInstSemiringCentroidHom {M : Type u_2} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Monoid M] [DistribMulAction M α] [SMulCommClass M α α] [IsScalarTower M α α] : DistribMulAction M (CentroidHom α)

Equations

• One or more equations did not get rendered due to their size.

instance CentroidHom.instModuleCentroidHomInstAddCommMonoidCentroidHom {R : Type u_4} {α : Type u_5} [NonUnitalNonAssocSemiring α] [Semiring R] [Module R α] [SMulCommClass R α α] [IsScalarTower R α α] : Module R (CentroidHom α)

Equations

• One or more equations did not get rendered due to their size.

theorem CentroidHom.centroid_eq_centralizer_mulLeftRight {α : Type u_5} [NonUnitalNonAssocSemiring α] : RingHom.rangeS (CentroidHom.toEndRingHom α) = Subsemiring.centralizer (Set.range ⇑AddMonoid.End.mulLeft ∪ Set.range ⇑AddMonoid.End.mulRight)

def CentroidHom.centerToCentroid {α : Type u_5} [NonUnitalNonAssocSemiring α] : ↥(NonUnitalSubsemiring.center α) →ₙ+* CentroidHom α

The canonical homomorphism from the center into the centroid

Equations

• One or more equations did not get rendered due to their size.

theorem CentroidHom.centerToCentroid_apply {α : Type u_5} [NonUnitalNonAssocSemiring α] (z : ↥(NonUnitalSubsemiring.center α)) (a : α) : (CentroidHom.centerToCentroid z) a = ↑z * a

theorem CentroidHom.center_iff_op_centroid {α : Type u_5} [NonUnitalNonAssocSemiring α] (a : α) : a ∈ NonUnitalSubsemiring.center α ↔ AddMonoid.End.mulLeft a = AddMonoid.End.mulRight a ∧ AddMonoid.End.mulLeft a ∈ Set.range CentroidHom.toEnd

def CentroidHom.centerIsoCentroid {α : Type u_5} [NonAssocSemiring α] : ↥(NonUnitalSubsemiring.center α) ≃+* CentroidHom α

The canonical isomorphism from the center of a (non-associative) semiring onto its centroid.

Equations

• One or more equations did not get rendered due to their size.

instance CentroidHom.instNegCentroidHomToNonUnitalNonAssocSemiring {α : Type u_5} [NonUnitalNonAssocRing α] : Neg (CentroidHom α)

Negation of CentroidHoms as a CentroidHom.

Equations

• One or more equations did not get rendered due to their size.

instance CentroidHom.instSubCentroidHomToNonUnitalNonAssocSemiring {α : Type u_5} [NonUnitalNonAssocRing α] : Sub (CentroidHom α)

Equations

• One or more equations did not get rendered due to their size.

instance CentroidHom.instIntCastCentroidHomToNonUnitalNonAssocSemiring {α : Type u_5} [NonUnitalNonAssocRing α] : IntCast (CentroidHom α)

Equations

• CentroidHom.instIntCastCentroidHomToNonUnitalNonAssocSemiring = { intCast := fun (z : ℤ) => z • 1 }

@[simp]

theorem CentroidHom.coe_int_cast {α : Type u_5} [NonUnitalNonAssocRing α] (z : ℤ) : ⇑↑z = z • ⇑(CentroidHom.id α)

theorem CentroidHom.int_cast_apply {α : Type u_5} [NonUnitalNonAssocRing α] (z : ℤ) (m : α) : ↑z m = z • m

@[simp]

theorem CentroidHom.toEnd_neg {α : Type u_5} [NonUnitalNonAssocRing α] (x : CentroidHom α) : CentroidHom.toEnd (-x) = -CentroidHom.toEnd x

@[simp]

theorem CentroidHom.toEnd_sub {α : Type u_5} [NonUnitalNonAssocRing α] (x : CentroidHom α) (y : CentroidHom α) : CentroidHom.toEnd (x - y) = CentroidHom.toEnd x - CentroidHom.toEnd y

instance CentroidHom.instAddCommGroupCentroidHomToNonUnitalNonAssocSemiring {α : Type u_5} [NonUnitalNonAssocRing α] : AddCommGroup (CentroidHom α)

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem CentroidHom.coe_neg {α : Type u_5} [NonUnitalNonAssocRing α] (f : CentroidHom α) : ⇑(-f) = -⇑f

@[simp]

theorem CentroidHom.coe_sub {α : Type u_5} [NonUnitalNonAssocRing α] (f : CentroidHom α) (g : CentroidHom α) : ⇑(f - g) = ⇑f - ⇑g

@[simp]

theorem CentroidHom.neg_apply {α : Type u_5} [NonUnitalNonAssocRing α] (f : CentroidHom α) (a : α) : (-f) a = -f a

@[simp]

theorem CentroidHom.sub_apply {α : Type u_5} [NonUnitalNonAssocRing α] (f : CentroidHom α) (g : CentroidHom α) (a : α) : (f - g) a = f a - g a

@[simp]

theorem CentroidHom.toEnd_int_cast {α : Type u_5} [NonUnitalNonAssocRing α] (z : ℤ) : CentroidHom.toEnd ↑z = ↑z

instance CentroidHom.instRing {α : Type u_5} [NonUnitalNonAssocRing α] : Ring (CentroidHom α)

Equations

• One or more equations did not get rendered due to their size.

@[reducible]

def CentroidHom.commRing {α : Type u_5} [NonUnitalRing α] (h : ∀ (a b : α), (∀ (r : α), a * r * b = 0) → a = 0 ∨ b = 0) : CommRing (CentroidHom α)

A prime associative ring has commutative centroid.

Equations

• CentroidHom.commRing h = let src := CentroidHom.instRing; CommRing.mk (_ : ∀ (f g : CentroidHom α), f * g = g * f)